On
damped
or
strongly damped hyperbolic
system
By
TAKEYUKI NAGASAWA (長澤壯之) ATSUSHI TACHIKAWA (立川 篤)
Mathematical Institute (Kawauchi) Department of Mathematics
Faculty ofScience, T\^ohoku University Faculty of Liberal Arts, Shizuoka University
Kawauchi, Aoba, Sendai, 980 836 Ohya, Shizuoka, 422
JAPAN JAPAN
1
Introduction
Let $\Omega$ be a bounded domain in $R^{k}$ with Lipschitz boundary
$\partial\Omega$. For a map
$u$ : $\Omega\cross$
$(0, \infty)arrow R^{t}$, we consider the hyperbolic system
$a:j(x)D_{t}^{2}u^{i}(x, t)-D_{\beta}(b_{lj}^{\alpha\beta}(x)D_{\alpha}u^{:}(x, t))+c_{ij}(x)||u(x, t)||_{c}^{m-2}u^{i}(x, t)$
(1.1)
$+a_{i_{J}}\cdot(x)D_{t}u^{i}(x, t)=0$ in $\Omega\cross(0, \infty),$ $j=1,$ $\cdots,$ $\ell$,
or
$a_{ij}(x)D_{t}^{2}u^{l}(x,t)-D_{\beta}(b_{ij}^{\alpha\beta}(x)D_{\alpha}u^{i}(x, t))+c_{ij}(x)||u(x,t)||_{c}^{m-2}u^{\mathfrak{i}}(x, t)$
(1.2)
- $D_{t}D_{\beta}(f_{lj}^{\alpha\beta}(x)D_{\alpha}u^{i}(x,t))=0$ in $\Omega\cross(0, \infty),$ $j=1,$ $\cdots,$ $\ell$.
Here $D_{t}$ and $D_{\alpha}$ mean the partial derivatives with respect to variable $t$ and $x^{\alpha},$ $i.e.$,
$D_{t}=\partial/\partial t$, $D_{\alpha}=\partial/\partial x^{a}$.
The c-norm $||\cdot||_{c}$ of $u$ is the square root of the quadratic form $c_{ij}u^{i}u^{j}$. Similar notations
$||u||_{a},$ $||Du||_{b}$ and
1
$Du||_{f}$ will appear later, and their meaning are$||u||_{a}=(a_{i_{J}}\cdot u^{i}u^{j})^{\frac{1}{2}}$ , $||Du||_{b}=(b_{ij}^{\alpha\beta}D_{\alpha}u^{i}D_{\beta}u^{J})^{\frac{1}{2}}$ , $||Du||_{f}=(f_{ij}^{\alpha\beta}D_{\alpha}u^{i}D_{\beta}u^{j})^{\frac{1}{2}}$ .
Here and in the following, summation over repeated indices is understood, the greek
in-dices runfrom 1 to $k$, and the latin ones from 1 to$\ell$. We assume that the coefficients $a_{j}(x)$,
$b_{ij}^{\alpha\beta}(x)$ and $c_{i_{\dot{J}}}(x)$ are bounded functions defined on $\Omega$ and satisfy the coercive condition
(1.3) $\{\begin{array}{l}a_{ij}(x)\xi^{i}\xi^{J}\geq\lambda_{0}|\xi|^{2}forall\xi\in R^{l}b_{ij}^{\alpha\beta}(x)\eta_{\alpha^{7}}^{i}/\beta\geq\lambda_{1}|\eta|^{2}forall\eta\in R^{k\ell}c_{ij}(x)\xi\cdot\xi^{j}\geq\lambda_{2}|\xi|^{2}forall\xi\in R^{t}f_{j}^{\alpha\beta}(x)\eta_{\alpha}\eta_{\beta}^{J}\geq\lambda_{3}|\eta|^{2}forall\eta\in R^{k\ell}\end{array}$
for some positive constants $\lambda_{0},$ $\lambda_{1},$ $\lambda_{2}$ and $\lambda_{3}$, and the symmetry
(1.4) $a_{ij}(x)=a_{ji}(x)$, $b_{ij}^{\alpha\beta}(x)=b_{j}^{\beta_{1}\alpha}(x)$, $c_{\mathfrak{i}j}(x)=c_{ji}(x)$, $f_{i_{J}^{\alpha\beta}}\cdot(x)=f_{ji}^{\beta\alpha}(x)$.
We call (1.1) the damped hyperbolic system, or the hyperbolic system with a damping term
$a_{1j}(x)D_{t}u^{i}(x, t)$. And the second system is called the strongly damped hyperbolic system,
or the hyperbolic system with a strongly damping $term-D_{t}D_{\beta}(f_{ij}^{\alpha\beta}(x)D_{\alpha}u^{i}(x, t))$. The
strongly damping term is also called the vescosity term. These system appear in some
models of continuum mechanics. For the historical remark we can refer [2] and references
cited therein.
We impose the initial and boundary conditions
(1.5) $u(x, 0)=u_{0}(x)$, $D_{t}u(x, 0)=v_{0}(x)$ in $\Omega$,
(1.6) $u(x, t)=w(x)$ on $\partial\Omega\cross(0, \infty)$,
where $u_{0}(x),$ $v_{0}(x)$ and $w(x)$ are given maps satisfying the compatibility condition $u_{0}(x)=$ $w(x)$ on $\partial\Omega$.
Our aim is two-folds. The first one is to construct global weak solutions by the method
of time-discretization. And the second one is to show their decay properly as $tarrow\infty$ in
case of$w\equiv 0,$ $i,e.$, homogeneous Dirichlet’s boundary condition. $F_{\dot{i}}rst$ we give the notion of weak solution. Let
$\gamma_{\partial\Omega}$ and $\gamma_{t=0}$ denote the trace operators to
$\partial\Omega$ and $\Omega\cross\{0\}$ respectively.
Definition
1.1. For $u_{0},$ $w\in H^{1,2}(\Omega)\cap L^{m}(\Omega)$ and $v_{0}\in L^{2}(\Omega)$ satisfying $\gamma_{\partial\Omega}u_{0}=\gamma_{\partial\Omega}w$,a map $u:\Omega\cross[0,T$) $arrow R^{t}$ is called a weak solution of (1.1) on $\Omega\cross[0, T$) with the initial
and boundary conditions $(1.5)-(1.6)$, if the following conditions are satisfied: (i) $u\in L^{\infty}(0, T;H^{1,2}(\Omega)\cap L^{m}(\Omega))$ with $D_{t}u\in L^{\infty}(0, T;L^{2}(\Omega))$.
(iii) For any $\psi(x,t)\in C_{0^{1}}([0, T);C_{0}(\Omega))\cap C([0, T);C^{1}(\Omega))$,
$\int_{0}^{T}\int_{\Omega}(-a_{tj}(x)D_{t}u^{i}(x, t)D_{t}\psi^{J}(x, t)+b_{ij}^{\alpha\beta}(x)D_{\alpha}u^{i}(x, t)D_{\beta}\psi^{J}(x, t)$
$+c_{ij}(x)||u(x, t)||_{c}^{m-2}u^{i}(x, t)\psi^{j}(x, t)+a:j(x)D_{t}u^{i}(x, t)\psi^{J}(x, t))dxdt$
$= \int_{\Omega}a_{ij}(x)v_{0}^{i}(x)\psi^{j}(x, 0)dx$ .
Definition
1.2. For $u_{0},$ $w\in H^{1,2}(\Omega)\cap L^{m}(\Omega)$ and $v_{0}\in H_{0}^{1,2}(\Omega)$ satisfying $\gamma_{\theta\Omega}u_{0}=\gamma_{\text{\^{o}}\Omega}w$,a map $u$ : $\Omega\cross[0, T$) $arrow R^{\ell}$ is called a weak solution of (1.2) on $\Omega\cross[0, T$) with the initial
and boundary conditions $(1.5)-(1.6)$, if the following conditions are satisfied:
(i) $u\in L^{\infty}(0, T;H^{1,2}(\Omega)\cap L^{m}(\Omega))$ with $D_{t}u\in L^{\infty}(0, T;L^{2}(\Omega))\cap L^{2}(0, T\cdot, H_{0}^{1,2}(\Omega))$.
(ii) $\gamma_{c=0}u(x, t)=u_{0}(x)$ and $\gamma_{\partial\Omega}u(x, t)=\gamma_{\partial\Omega}w(x)$ for
$0<t<T$
.(iii) For any $\psi(x, t)\in C_{0}^{1}([0, T);C_{0}(\Omega))\cap C([0, T);C^{1}(\Omega))$,
$\int_{0}^{T}\int_{\Omega}(-a_{lj}(x)D_{t}u^{l}(x, t)D_{t}\psi(x, t)+b_{ij}^{\alpha\beta}(x)D_{\alpha}u^{i}(x, t)D_{\beta}\psi^{j}(x, t)$
$+c_{lj}(x)||u(x, t)||_{c}^{m-}u^{i}(x,t)\psi^{j}(x, t)+f_{ij}^{\alpha\beta}(x)D_{t}D_{\alpha}u^{i}(x, t)D_{\beta}\psi(x, t))dxdt$
$= \int_{\Omega}a_{ij}(x)v_{0}^{1}(x)\psi^{j}(x, 0)dx$.
Definition
1.3. We say $u$ is a global weak solution if $u|_{\Omega\cross[0,T)}$ is a weak solution on $\Omega\cross[0, T)$ for any $T>0$.We discuss the damped and strongly damped hyperbolic systems in
\S
2 and\S
3 respec-tively. This note is an epitome of $[5, 6]$.2
The damped
hyperbolic
system
2.1
A
construction
of
weak solutions
Here we construct weak solutions by use of a combination of time-discretization and calculus of variations. Though our system solved in several different way, we omit the
historical remark of the equations. The authors, however, think that our method is not
equations. The method explained here was firstly introduced by Rektorys [7] in 1971. He applied it to linear parabolic equations. Independently Kikuchi [3] rediscovered this method in 1991, and he tried to apply the method to non-linear equations coming from variational problems. Actually Bethuel-Coron-Ghidaglia-Soyeur [1] constructed a weak solution of the heat flow for harmonic maps by the method. The authors also constructed weak solutions of a semi-linear hyperbolic system and the Navier-Stokes equations by the method in [9] and [4] respectively.
We firstly construct an approximate solution as follows. Let $h$ be a positive number, which will tend to zero later. $u_{0}$ is a given initial data of $u$. $u_{1}$ is defined by
$u_{1}(x)=u_{0}(x)+v(x, h)$,
where $v$ is an $R^{\ell}$-valued function satisfying
(2.7) $\{\begin{array}{l}v(x,0)=0,D_{t}v(x,0)=v_{0}(x)in\Omega v(x,t)=0on\partial\Omega\cross Rv\in L^{\infty}(R\cdot.H^{1,2}(\Omega)\cap L^{m}(\Omega))D_{t}v(\cdot,t)isaweak1ycontinuousmapoftwithva1uesinL^{2}(\Omega)\int_{\Omega}(\frac{1}{2}|D_{t}v|^{2}+\frac{1}{2}||Dv^{i}||^{2}+\frac{1}{m}|v|^{m})dx\leq\int_{\Omega}\frac{1}{2}|v_{0}^{i}|^{2}dx\end{array}$
Here $||\cdot||$ denotes the Euclidean norm, and $D=(D_{1}, \cdots, D_{k})$. To get such a map $v$, for
example, we solve the initial-boundary value problem
(2.8) $\{\begin{array}{l}D_{t}^{2}v(x,t)-\triangle v(x,t)+|v^{i}|^{m-2}v^{i}(x,t)=0v^{i}(x,0)=0,D_{t}v^{i}(x,0)=v_{\dot{0}}(x)v(x,t)=0\end{array}$ $inonon\Omega^{\Omega}\partial\Omega^{\cross}x^{R}R$
.
[8, Theorem 2] guarantees the existence of weak solution $v$ satisfying (2.7).
For $n\geq 2$ we define $u_{n}$ as a minimizer ofthe functional
$\mathcal{F}_{n}(u)=\int_{\Omega}(\frac{1}{2}\frac{||u-2u_{n-1}+u_{n-2}||_{a}^{2}}{h^{2}}+\frac{1}{2}||Du||_{b}^{2}+\frac{1}{m}||u||_{c}^{m}+\frac{1}{2}\frac{[|u-u_{n-2}||_{a}^{2}}{2h})dx$
in the class
$\{u\in H^{1,2}(\Omega)\cap L^{m}(\Omega) ; \gamma_{\partial\Omega}u=\gamma_{\partial\Omega}w\}$.
The functional $\mathcal{F}_{n}(u)$ is coercive in the above class, and the standard argument of
Proposition 2.1. $\mathcal{F}_{n}(u)$ has a minimizer, which we denote by
$u_{n}$. It
satisfies
theEuler-Lagrange equation
$0= \frac{d}{d\epsilon}\mathcal{F}_{n}(u+\epsilon\varphi)|_{\epsilon=0}$
(2.9) $= \int_{\Omega}\{\frac{1}{h^{2}}a_{ij}(x)(u^{i}-2u_{n-1}+u_{n-2}^{i})\psi+b_{ij}^{\alpha\beta}(x)D_{\alpha}u^{i}D_{\beta}\varphi^{l}+c_{ij}(x)||u||_{c}^{m-2}u\varphi$
$+ \frac{1}{2h}a_{ij}(x)(u‘ -u_{n-2}^{i})\varphi\}dx$
for
all $\varphi\in H_{0}^{1,2}(\Omega)\cap L^{m}(\Omega)$.Thus $\{u_{n}\}$ is well-defined inductively. Now, using $\{u_{n}\}$, we define two maps
$u_{h}$ and $\overline{u}_{h}$
by
$\{\begin{array}{l}\overline{u}_{h}(x,t)=\{u_{h}(x,t)=\{u_{0}(x)+v(x,t)_{n}\frac{t-(n-1)h}{h}u(x)+\frac{nh-t}{h}u_{n-1}(x)\end{array}$
$forfor$ $(n-1)h<t\leq nh-1\leq t\leq h$
,
$n\geq 2$.
They approximate a weak solution of (1.1).
Proposition 2.2. For small$h\in(0,1)$ it holds that
$\{\begin{array}{l}\{\overline{u}_{h}\},\{u_{h}\}areboundedsetinL^{m’}(\Omega\cross(0,T)),wherem^{/}=\max\{2,m\}\{D_{t}u_{h}\}isaboundedsetinL^{2}(\Omega\cross(0,T))\cap L^{\infty}(0,T\cdot.L^{2}(\Omega))\{D_{\alpha}\overline{u}_{h}\},\{D_{\alpha}u_{h}\}areboundedsetinL^{2}(\Omega\cross(0,T))\end{array}$
and
$\int_{0}^{T}\int_{\Omega}|\overline{u}_{h}-u_{h}|^{2}dxdt=O(h^{2}T)$.
Sketch
of Proof.
Since $u_{n}$ and $u_{n-2}$ coincide on $\partial\Omega,$ $u_{n}-u_{n-2}(n\geq 2)$ is an admissibleIt follows from Propositions 2.1 and 2.2 that
$\int_{0}^{T}\int_{\Omega}\{\frac{1}{h}a_{j}(x)(D_{t}u_{h}^{\mathfrak{i}}(x, t)-D_{t}u_{h}(x, t-h))\psi(x)$
$+b_{ij}^{\alpha\beta}(x)D_{\alpha}\overline{u}_{h}^{i}(x, t)D_{\beta}\varphi’(x)+c_{lj}(x)||\overline{u}_{h}(x, t)||_{c}^{m-2}\overline{u}_{h}^{i}(x, t)\psi(x)$
$+ \frac{1}{2}a_{lj}(x)(D_{t}u_{h}^{*}(x, t)+D_{t}u_{h}^{i}(x, t-h))\psi(x)\}\eta(t)dxdt$
$=0(1)$ as $h\downarrow 0$
for any $T>0$ and $\eta\in C_{0^{\infty}}[0, T$).
The weak(-star) compactness argument and the diagonal argument give the fact that $\overline{u}_{h}$
and $u_{h}$ converge to a global weak solution $u$ along a suitable subsequence of $h\downarrow 0$. Thus
we get the following result.
Theorem 2.1. Let $m>1$. For any $u_{0)}w\in H^{1,2}(\Omega)\cap L^{m}(\Omega)$ and $v_{0}\in L^{2}(\Omega)$ satisfying
$\gamma_{\partial\Omega}u_{0}=\gamma_{\theta\Omega}w$, there exists at least one global weak solution $u$ to (1.1), (1.5) and (1.6).
2.2
Decay of
our
weak
solutions
In this subsection we assume $w\equiv 0$ and $m\geq 2$.
Since we are posing the homogeneous boundary condition, $u_{n}$ is an admissible test
func-tion for (2.9). Therefore we can see that
$\int_{\Omega}\frac{1}{h^{2}}a_{1j}(u_{n}^{i}-u_{n-1}^{i})(u_{n-1}^{j}-u_{n-2}^{j})dx$
$= \int_{\Omega}\{\frac{1}{h}(a_{\dot{j}}u_{\dot{n}}\frac{u_{n}^{j}-u_{n-1}^{j}}{h}-a_{ij}u_{n-1}^{i}\frac{u_{n-1}^{j}-u_{n-2}^{j}}{h})+a_{\dot{j}}u_{n}^{i}\frac{u_{n}^{J}-u_{n-1}^{j}}{h}$
Next we test (2.9) by $\varphi=u_{n}-u_{n-1}$ to get $0= \int_{\Omega}[\frac{1}{h^{2}}a_{ij}\{(u_{n}^{i}-u_{n-1})-(u_{n-1}^{i}-u_{n-2}^{i})\}(u_{n}^{\dot{J}}-u_{n-1}^{j})$ $+b_{\dot{j}}^{\alpha\beta}D_{\alpha}u_{n}:(D_{\beta}u_{n}^{j}-D_{\beta}u_{\dot{n}-1}^{J})+c_{ij}||u_{n}||_{c}^{m-2}u_{n}^{l}(u_{n}^{j}-u_{n-1}^{j})$ $+ \frac{1}{2h}a_{\mathfrak{i}j}(u_{n}^{i}-u_{n-2}^{i})(u_{n}^{j}-u_{n-1}^{j})]dx$ $= \int_{\Omega}[\frac{1}{h^{2}}\{||u_{n}-u_{n-1}||_{a}^{2}-a_{\dot{J}}(u_{n-1}^{i}-u_{n-2}^{\dot{l}})(u_{n}^{j}-u_{n-1}^{\dot{J}})\}$ $+||Du_{n}||_{b}^{2}-b_{ij}^{\alpha\beta}D_{\alpha}u_{n}^{i}D_{\beta}u_{n-1}^{j}+||u_{n}||_{c}^{m}-||u_{n}||_{c}^{m-2}c_{lj}u_{n}^{i}u_{n-1}^{j}$ $+ \frac{1}{2h}||u_{n}-u_{n-1}||_{a}^{2}+\frac{1}{2h}a_{i_{J}}(u_{n}^{:}-u_{n-1}^{i})(u_{n-1}^{\dot{J}}-u_{n-2}^{\dot{J}})]dx$.
Combining theserelations, and estimating non-coercive terms by use of Young’s inequality,
we get
Proposition 2.3. It holds that
$\frac{\Psi_{h}(t)-\Psi_{h}(t-h)}{h}+\Psi_{h}(t)\leq hK_{1}$,
where
$\Psi_{h}(t)=\int_{\Omega}(\frac{1}{2}||D_{t}u_{h}||_{a}^{2}+\frac{1}{2}a:_{\dot{J}}\overline{u}_{h}^{1}D_{t}u_{h}^{j}+\frac{1}{2}||D\overline{u}_{h}||_{b}^{2}+\frac{1}{m}||\overline{u}_{h}||_{c}^{m})dx$,
and $K_{1}$ is a constant depending on the initial data but not on $h$. And
therefore
we have$\Psi_{h}(t)\leq(\frac{1}{1+h})^{n}\Psi_{h}(+0)+hK_{1}$,
where the relation between $t$ and $n$ is given by
$n=\lceil t/h\rceil$ ,
$\lceil\rceil$ denotes the ceiling, $i.e.,$ $\lceil x\rceil$ is the smallest integer greater than or equal to $x$.
Passing to $h\downarrow 0$, we have
for almost every $t\in(0, \infty)$. Since $u$ is a weak solution, it belongs to $C([0, T];L^{2}(\Omega))$ and
$D_{t}u$ to $L^{\infty}(0, T;L^{2}(\Omega))$. Hence it follows from the above differential inequality that
$||u(\cdot,t)||_{L^{2}(\Omega)}^{2}\leq K_{3}e^{-C_{2}t}$. Using $\Psi_{h}$ again, we have
$\int_{\Omega}(\frac{1}{4}||D_{t}u_{h}||_{a}^{2}+\frac{1}{2}||D\overline{u}_{h}||_{b}^{2}+\frac{1}{m}||\overline{u}_{h}||_{c}^{m})dx\leq(\frac{1}{1+h})^{n}K_{4}+hI\zeta_{1}+C_{3}||\overline{u}_{h}(\cdot, t)||_{L^{2}(\Omega)}^{2}$.
Passing to $h\downarrow 0$ again, we obtain
Theorem 2.2. Our weak solution
satisfies
$||u(\cdot, t)||_{H^{1,2}(\Omega)}^{2}+||u(\cdot, t)||_{L(\Omega)}^{m_{m}}+||D_{t}u(\cdot, t)||_{L^{2}(\Omega)}^{2}\leq Ke^{-Ct}$,
provided $w\equiv 0$ and $m\leq 2$.
Remark 2.1. If we define $u_{n}(n\geq 2)$ as a minimizer of
$\tilde{\mathcal{F}}_{n}(u)=\int_{\Omega}(\frac{1}{2}\frac{||u-2u_{n-1}+u_{n-2}||_{a}^{2}}{h^{2}}+\frac{1}{2}||Du||_{b}^{2}+\frac{1}{m}||u||_{c}^{m}+\frac{1}{2}\frac{||u-u_{n-1}||_{a}^{2}}{h})dx$
instead of $\mathcal{F}_{n}(u)$, then we can also construct a global weak solution form them. The
authors, however, cannot clarify that this weak solution has a decay property or not.
3
The strongly damped hyperbolic system
In this section we consider the initial-boundary value problem for the strongly hyperbolic system (1.2). The method is similar as in
\S
2, so we state only the scheme. Let $h$ be a positive number, which will tend to zero later. $u_{0}$ is a given initial data of $u$. $u_{1}$ is definedby
$u_{1}(x)=u_{0}(x)+hv_{0}(x)$,
where $v_{0}$ is $al$so given initial data of $D_{t}u$. For $n\geq 2$ we define $u_{n}$ as a minimizer of the
functional
$\mathcal{G}_{n}(u)=\int_{\Omega}(\frac{1}{2}\frac{||u-2u_{n-1}+u_{n-2}||_{a}^{2}}{h^{2}}+\frac{1}{2}||D$川$|_{b}^{2}+ \frac{1}{m}$
||
川に $+ \frac{1}{2}\frac{||D(u-u_{n-1})||_{f}^{2}}{h}I^{dx}$
in the class
This scheme gives us
Theorem 3.1. Let$m>1$. For any$u_{0;}w\in H^{1,2}(\Omega)\cap L^{m}(\Omega)$ and$v_{0}\in H_{0}^{1,2}(\Omega)$ satisfying
$\gamma_{\partial\Omega}u_{0}=\gamma_{\partial\Omega}w$, there exists at least one global weak solution $u$ to (1.2), (1.5) and (1.6).
If
$w\equiv 0$, then our weak solutionsatisfies
$||u(\cdot, t)||_{H^{1,2}(\Omega)}^{2}+||u(\cdot, t)||_{L(\Omega)}^{m_{m}}+||D_{t}u(\cdot, t)||_{L^{2}(\Omega)}^{2}\leq Ke^{-Ct}$.
Remark 2.1. If we define $u_{n}(n\geq 2)$ as a minimizer of
$\tilde{\mathcal{G}}_{n}(u)=\int_{\Omega}(\frac{1}{2}\frac{||u-2u_{n-1}+u_{n-2}||_{a}^{2}}{h^{2}}+\frac{1}{2}||Du||_{b}^{2}+\frac{1}{m}||u||_{c}^{m}+\frac{1}{2}\frac{||D(u-u_{n-2})||_{f}^{2}}{2h})dx$
instead of$\mathcal{G}_{n}(u)$, then we can alsoconstruct aglobalweak solutionformthem. The authors,
however, cannot clarify that this weak solution has a decay property or not.
A technical difference between $\mathcal{F}_{n}(u)$ and $\tilde{\mathcal{F}}_{n}(u)$ (see Remark 2.1), and between $\mathcal{G}_{n}(u)$
and $\tilde{\mathcal{G}}_{n}(u)$ comes form Poincare’s inequality. The inequality can be expressed by $(u, u)_{L^{2}(\Omega)}\leq C(Du, Du)_{L^{2}(\Omega)}$
for $u\in H_{0}^{1,2}(\Omega)$. It, however, does not hold that
$(u_{n}, u_{n-1})_{L^{2}(\Omega)}\leq C(Du_{n}, Du_{n-1})_{L^{2}(\Omega)}$.
We must choose scheme so that such terms does not appear incalculating adecay property.
References
[1] Bethuel, F., J.-M. Coron, J.-M. Ghidaglia&A. Soyeur, Heat
flows
and relaxed energiesfor
harmonic maps, in “Nonlinear Diffusion Equations and Their Equilibrium States, 3”, ed.: N. G. Lloyd, W. M. Ni, L. A. Peletier, J. Serrin, Progr. Nonlinear DifferentialEquations Appl. 7, Birkh\"auser, Boston . Basel . Berlin, 1992, pp. 99-109.
[2] Kawashima, S.
&Y.
Shibata, Global existence and exponential stabilityof
small solu-tions to nonlinear viscoelasticity, Comm. Math. Phys. 148 (1992), 189-208.[3] Kikuchi, N., An approach to the construction
of
Morseflows for
variationalfunc-tionals, in ((
$Nematics$ Mathematical and Physical Aspects”, ed.: M. Coron,
J.-M. Ghidaglia&F. H\’elein, NATO Adv. Sci. Inst. Ser. $C$: Math. Phys. Sci. 332, Kluwer
[4] Nagasawa, T., Construction
of
weak solutionsof
the Navier-Stokes equations on Rie-mannianmanifold
by minimizing variational functionals, preprint.[5] Nagasawa, T.
&A.
Tachikawa, Existence and asymptotic behaviorof
weak solutions to semilinear hyperbolic systems with damping term, preprint.[6] Nagasawa, T.
&A.
Tachikawa, Existence and asymptotic behaviorof
weak solutions to strongly damped semilinear hyperbolic system, preprint.[7] Rektorys, K., On application
of
direct variational methods to the solutionof
parabolic boundary value problemsof
arbitrary order in the space variables, Czechoslovak Math.J. 21 (1971), 318-339.
[8] Strauss, W., On weak solutions
of
semi-linear hyperbolic equations, An. Acad. Brasil.Ci\^enc. 42 (1970), 645-651.
[9] Tachikawa, A., A variational approach to constructing weak solutions
